Redoing Tetration.org

[Daniel]

I am rewriting my Tetration.org website, which is now twenty years old. There are a number of questions asked over and over again here, on MathOverflow.net and Math.stackexchange.com that I want to address on my website.

A few fundamental questions follow:

• What types of tetration extensions are there?
  
• What is the "best" tetration extension?
  
• To what degree has the issue of convergence been settled for tetration?
  
• What types of hyperoperator extensions are there?
  
• What is the most general form of \(a\uparrow ^n b\)? Can \(n\) take real and complex values?
  

What other fundamental questions should I deal with?

[marracco]

Tetration of real variable should not be considered a single valued function. It leaves out the quasi periodic, square wave behavior, the multiple asymptotes. It at least has 5 branches.

[Daniel]

Tetration of real variable should not be considered a single valued function. It leaves out the quasi periodic, square wave behavior, the multiple asymptotes. It at least has 5 branches.

Thanks for the feedback. I think I know what you are talking about, but it would be nice if you could expand on your answer.

[marracco]

Tetration of real variable should not be considered a single valued function. It leaves out the quasi periodic, square wave behavior, the multiple asymptotes. It at least has 5 branches.

Thanks for the feedback. I think I know what you are talking about, but it would be nice if you could expand on your answer.

Take a base \(1 < b \leq e^{e^{-1}}\), for example, \(b=\sqrt{2}\)

The exponential function \(y_(x)=b^x\) has 5 parts:

1. \(y < 2\)
   
2. \(y= 2\) this is a fixed point
   
3. \(2<y<4\)
   
4. \(y= 4\) this is a fixed point
   
5. \(y>4\)
   

1. If you start iterating tetration at \(y_0=1\) (or any \(y_0<2\)), you get the standard tetration, that reaches the asymptote at the lower fixpoint, given by the lambert \(w\) function, as is known. Let's call this the branch 1 or main branch.
   
2. If you start iterating tetration at \(y_0=2\) this is a fixed point, so you stay in the asymptote given by the \(w\) function. This is the branch 2, which is an asymptote at  \(y=\frac{w(-ln(b))}{-ln(b)}=2\).
   
3. \(2<y_0<4\) gives a Z curve, which starts at \(^{-\infty}b=4\) (the asymptote \(y=4\)), and ends at  \(^{\infty}b=2\), given by the \(w\) function.
   
4. \(y_0= 4\) this is a fixed point, and another asymptote.
   
5. \(y_0>4\) Is a super exponential function.
   

Now, for bases \(0 \leq b <e^{-e}\), the tetration function oscillates, between 2 asymptotes, rapidly converging to a square wave function.


The asymptotes and the center are the 3 solutions \(c\) of \(b^{c^c}=c\).

Those asymptotes must be related to the other asymptotes for bases \(b>1\) in some way. There must exist some transformation that converts the asymptotes, and the Z curve must convert into the part of the curve that switches between asymptotes. The main branch must convert into the upper side of the square wave, and the upper super exponential branch must be transformed into the lower side of the square wave.

Maybe tetration introduces transfinite numbers which turn the main branch into a square wave function at the transfinite scale, and we only see the real-number part commonly graphed as the main branch.